Academic literature on the topic 'Satellite orbital motion model'

The present study concerns the determination of ocean tide model parameters from GOCE orbital perturbation analysis. The GOCE satellite was launched by the European Space Agency in 2009 and is flying on a Sun-synchronous near circular orbit, at the very low altitude of about 250 km which makes it very sensitive to tidally induced orbit perturbations. The strategy adopted for analyzing GOCE GPS tracking data is the direct fully-dynamic approach, consisting in the GOCE precise orbit determination (POD) and accumulation of the normal equations for each orbital arc, followed by a multiarc solution for the estimation of the global ocean tide parameters. The GOCE GPS observations are processed using the NAPEOS S/W system (ESA/ESOC), specific for satellite orbit determination and prediction, upgraded to inclusion of the partial derivatives with respect to the ocean tide parameters and the ocean tide model inversion capability. A sensitivity study of the ocean tide perturbations on GOCE orbit was carried out using as a reference the FES2004 model, in order to define the set of tidal harmonic parameters affecting GOCE orbit. In particular, the secular rates of the GOCE angular elements are estimated through a linear least-square fit. From GOCE mean orbital characteristics, the spectral analysis of ocean tide perturbations in the radial, transverse and normal direction is performed using Kaula's linear satellite theory. Then, the perturbation statistics by coefficient is computed, obtaining a maximum RMS of about 1.323 m for the radial component, 363.136 m for the transverse component and 76.241 m for the normal component. The temporal aliasing problem is also accounted for the recovery of tidal parameters with GOCE and the principal alias periods are calculated for each tidal perturbation frequency, considering the length of the available GOCE data record. To fix a limit for the number of parameters to be estimated, three different cutoffs are applied to the RMS perturbation coefficients, respectively equal to 5 mm for the radial component, 2 cm for the transverse component and 1 cm for the normal component, both in the prograde and retrograde case. The total parameters to be estimated result to be 490. GOCE data are processed to perform the fully-dynamic POD over daily orbital arcs from the 1st November 2009 until the 31st May 2011, but only arcs with a post-fit RMS of the GPS phase observations residuals lower than 8 mm are considered for the multiarc processing, for a total of 431 days. The obtained preliminary results show the relative error of the estimated parameters with respect to the corresponding FES2004 parameters lower than 1 for about the 16\% of the total, meaning that they are of the order of magnitude of the FES2004 parameters. GOCE orbital data were reprocessed along the same period of the previous run, initializing the ocean tide model with the estimated parameters, if present, and maintaining otherwise the FES2004 parameters. The post-fit RMS of the GPS phase residuals obtained with the new ocean tide model has a mean value of 6.5 mm, and it is noteworthy that the difference between the post-fit RMS obtained with the FES2004 model and that resulting from the new ocean tide model indicates a mean improvement of about 0.6 mm in for the 96\% of the analyzed arcs and greater than 1 mm for the 16\%, few days reach a difference of 2 mm. Finally, the orbits obtained with the estimated parameters are compared with the orbits obtained employing the FES2004 model and the official GOCE Reduced-Dynamic PSO. The 3D RMS of the difference between the orbits computed using FES2004 and those recomputed with the new parameters shows a mean value of 2.5 cm, while the 3D RMS of the difference with respect to the official R/D PSO has a mean value of 4.9 cm. Moreover, the difference between the 3D RMS of the orbit residuals between the R/D PSO and the GOCE POD with FES2004 and the RMS of the difference between the GOCE R/D PSO and the GOCE POD with the new parameters results to have a mean improvement of 0.9 cm. Further POD-Multiarc runs are certainly necessary, together with the refinement of the list of parameters to be estimated, removing excessively ill-estimated ocean tide parameters and introducing new parameters where appropriate. Indeed, the model parameter tuning and investigation is essential to adjust the best combination of parameters to be estimated. Moreover, an extension of the data set to much longer time-period should allow a substantial improvement of the obtained results. The task has proven very intensive and challenging, but the partial results obtained are encouraging and a motivation for future analysis.<br>Il presente lavoro di ricerca riguarda la determinazione dei parametri del modello di marea oceanica dall'analisi delle perturbazioni orbitali di GOCE. Il satellite GOCE è stato lanciato dall'Agenzia Spaziale Europea nel 2009 e volando a quota estremamente bassa, pari a circa 250 km, è molto sensibile alle perturbazioni orbitali indotte dalle maree. La strategia adottata per l'analisi dei dati GPS GOCE è l'approccio numerico diretto caratterizzato da un modello di forza completo, per questo detto completamente dinamico, che consiste nella determinazione orbitale precisa (POD) di GOCE con accumulo delle equazioni normali per ogni arco orbitale, seguiti da una soluzione multiarco per la stima dei parametri globali di marea oceanica. Le osservazioni GPS di GOCE vengono elaborate utilizzando il S/W NAPEOS (ESA/ESOC), specifico per la determinazione orbitale di satelliti e aggiornato per includere le derivate parziali rispetto ai parametri di marea e la struttura che permetta l'inversione del modello di marea. Uno studio di sensibilità delle perturbazioni mareali sull'orbita di GOCE è stato eseguito utilizzando come modello di riferimento il FES2004, al fine di definire la griglia di parametri di marea che influenzano maggiormente l'orbita di GOCE. In particolare, è stata seguita la seguente procedura. Prima di tutto, le variazioni secolari degli elementi angolari di GOCE (argomento di perigeo, longitudine del nodo ascendente, anomalia media) sono state stimate con un fit lineare ai minimi quadrati. Dalle caratteristiche orbitali medie di GOCE, è stata eseguita l'analisi spettrale delle perturbazioni di marea oceanica in direzione radiale, trasversale e normale, utilizzando la teoria lineare di Kaula. In seguito, è stata effettuata la statistica delle perturbazioni per coefficiente, ottenendo un RMS massimo di circa 1.323 m per la componente radiale, 363.136 m per la componente trasversale e 76.241 m per la componente normale. E' stato affrontato anche il problema di aliasing temporale di cui soffrono i parametri di marea che devono essere stimati con GOCE e per ogni frequenza di perturbazione mareale sono stati calcolati i periodi principali di aliasing; si è considerata infine la lunghezza del set di dati GOCE disponibili. E' stato inoltre necessario fissare un limite per il numero di parametri da stimare, tre soglie diverse sono state applicate all'RMS delle perturbazioni per coefficiente, rispettivamente pari a 5 mm per la componente radiale, 2 cm per la componente trasversale e 1 cm per la componente normale, sia nel caso progrado che retrogrado. In tal modo, i parametri totali da stimare risultano essere 490. I dati orbitali di GOCE sono stati analizzati per la stima della POD su archi orbitali giornalieri, coprendo un periodo che va dal 1 novembre 2009 al 31 maggio 2011, ma sono stati considerati per il multiarc solo i giorni con un post-fit RMS dei residui delle osservazioni di fase GPS inferiore a 8 mm, per un totale di 431 giorni. I risultati ottenuti sono preliminari e mostrano un errore relativo dei parametri stimati rispetto ai corrispondenti parametri del FES2004 inferiore a 1 per circa il 16% del totale, il che significa che sono dell'ordine di grandezza dei parametri del FES2004. I dati orbitali di GOCE sono stati poi rielaborati lungo lo stesso periodo di analisi, inizializzando il modello di marea oceanica con i nuovi parametri stimati, dove possibile, mantenendo altrimenti i parametri del FES2004. Il post-fit RMS dei residui di fase GPS ottenuti con il nuovo modello di marea ha un valore medio di 6.5 mm, ed è da notare che la differenza tra i post-fit RMS ottenuti con il FES2004 e quelli risultanti dal nuovo modello indicano un miglioramento medio di circa 0.6 mm per il 96% degli archi analizzati e maggiore di 1 mm per il 16%, mentre pochi giorni raggiungono una differenza di 2 mm. Infine, le orbite di GOCE ottenute con i parametri stimati vengono confrontate con le orbite ottenute usando il FES2004 e con le PSO ufficiali a dinamica ridotta. L'RMS 3D della differenza tra le orbite calcolate utilizzando il FES2004 e quelle calcolate con i nuovi parametri mostra un valore medio di 2.5 cm, mentre l'RMS 3D della differenza rispetto alle PSO a dinamica ridotta ha un valore medio di 4.9 cm. Inoltre, la differenza tra l'RMS 3D dei residui orbitali tra le PSO e la POD eseguita con il FES2004 e l'RMS 3D della differenza tra le PSO e la POD di GOCE eseguita con l'aggiunta dei parametri stimati mostra miglioramento medio di 0.9 cm. Ulteriori run di POD e multiarc sono certamente necessari, insieme alla rifinitura della lista dei parametri da stimare, rimuovendo quelli eccessivamente fuori dalla soluzione del FES2004 ed eventualmente introducendone opportunamente di nuovi. Infatti, è essenziale fare ulteriori e approfondite indagini per individuare la migliore combinazione di parametri di marea da stimare. Inoltre, l'estensione dei dati GOCE a un periodo di tempo più lungo dovrebbe consentire un sostanziale miglioramento dei risultati ottenuti. Il compito del presente lavoro di ricerca si è dimostrato molto intenso e impegnativo, ma i risultati parziali ottenuti sono incoraggianti e rappresentano una motivazione per le analisi future.

Spacecraft relative motion modeling and control promises to enable or augment a wide range of missions for scientific research, military applications, and space situational awareness. This dissertation focuses on the development of novel, optimization-based, control design for some representative relative-motion-enabled missions. Spacecraft relative motion refers to two (or more) satellites in nearly identical orbits. We examine control design for relative configurations on the scale of meters (for the purposes of proximity operations) as well as on the scale of tens of kilometers (representative of science gathering missions). Realistic control design for satellites is limited by accurate modeling of the relative orbital perturbations as well as the highly constrained nature of most space systems. We present solutions to several types of optimal orbital maneuvers using a variety of different, realistic assumptions based on the maneuver objectives. Initially, we assume a perfectly circular orbit with a perfectly spherical Earth and analytically solve the under-actuated, minimum-energy, optimal transfer using techniques from optimal control and linear operator theory. The resulting open-loop control law is guaranteed to be a global optimum. Then, recognizing that very few, if any, orbits are truly circular, the optimal transfer problem is generalized to the elliptical linear and nonlinear systems which describe the relative motion. Solution of the minimum energy transfer for both the linear and nonlinear systems reveals that the resulting trajectories are nearly identical, implying that the nonlinearity has little effect on the relative motion. A continuous-time, nonlinear, sliding mode controller which tracks the linear trajectory in the presence of a higher fidelity orbit model shows that the closed-loop system is both asymptotically stable and robust to disturbances and un-modeled dynamics. Next, a novel method of computing discrete-time, multi-revolution, finite-thrust, fuel-optimal, relative orbit transfers near an elliptical, perturbed orbit is presented. The optimal control problem is based on the classical, continuous-time, fuel-optimization problem from calculus of variations, and we present the discrete-time analogue of this problem using a transcription-based method. The resulting linear program guarantees a global optimum in terms of fuel consumption, and we validate the results using classical impulsive orbit transfer theory. The new method is shown to converge to classical impulsive orbit transfer theory in the limit that the duration of the zero-order hold discretization approaches zero and the time horizon extends to infinity. Then the fuel/time optimal control problem is solved using a hybrid approach which uses a linear program to solve the fuel optimization, and a genetic algorithm to find the minimizing time-of-flight. The method developed in this work allows mission planners to determine the feasibility for realistic spacecraft and motion models. Proximity operations for robotic inspection have the potential to aid manned and unmanned systems in space situational awareness and contingency planning in the event of emergency. A potential limiting factor is the large number of constraints imposed on the inspector vehicle due to collision avoidance constraints and limited power and computational resources. We examine this problem and present a solution to the coupled orbit and attitude control problem using model predictive control. This control technique allows state and control constraints to be encoded as a mathematical program which is solved on-line. We present a new thruster constraint which models the minimum-impulse bit as a semi-continuous variable, resulting in a mixed-integer program. The new model, while computationally more expensive, is shown to be more fuel-efficient than a sub-optimal approximation. The result is a fuel efficient, trajectory tracking, model predictive controller with a linear-quadratic attitude regulator which tracks along a pre-computed ``safe'' trajectory in the presence of un-modeled dynamics on a higher fidelity orbital and attitude model.<br>Ph. D.

Approved for public release; distribution is unlimited<br>The Naval Space Surveillance Center (NAVSPASUR) uses an analytic satellite motion model based on the Brouwer-Lyddane theory to assist in tracking over 6000 objects in orbit around the earth. The satellite motion model is implemented by a Fortran subroutine, PPT2. Due to the increasing number of objects required to be tracked, NAVSPASUR desires a method to reduce the computation time of this satellite motion model. Parallel computing offers one method to achieve this objective. This thesis investigates the parallel computing potential of the MAVSPASUR model using the Intel iPSC/2 hypercube multi-computer. The thesis developed several parallel algorithms for the NAVSPASUR satellite motion model using the various methods of parallelization, applies these algorithms to the hypercube, and reports on each algorithm's potential reduction in computation time. A diskette containing the Fortran software is available upon request from neta@boris.math.nps.navy.mil.

В работе исследуется возмущенное движение относительно неподвижной точки динамически симметричного тяжелого твердого тела под действием момента сил произвольной природы. Момент силы тяжести не рассматривается как возмущающий момент, а относится к невозмущенному движению, которое представляет собой движение в случае Лагранжа.

This thesis addresses visual tracking of a non-cooperative as well as a partially cooperative satellite, to enable close-range rendezvous between a servicer and a target satellite. Visual tracking and estimation of relative motion between a servicer and a target satellite are critical abilities for rendezvous and proximity operation such as repairing and deorbiting. For this purpose, Lidar has been widely employed in cooperative rendezvous and docking missions. Despite its robustness to harsh space illumination, Lidar has high weight and rotating parts and consumes more power, thus undermines the stringent requirements of a satellite design. On the other hand, inexpensive on-board cameras can provide an effective solution, working at a wide range of distances. However, conditions of space lighting are particularly challenging for image based tracking algorithms, because of the direct sunlight exposure, and due to the glossy surface of the satellite that creates strong reflection and image saturation, which leads to difficulties in tracking procedures. In order to address these difficulties, the relevant literature is examined in the fields of computer vision, and satellite rendezvous and docking. Two classes of problems are identified and relevant solutions, implemented on a standard computer are provided. Firstly, in the absence of a geometric model of the satellite, the thesis presents a robust feature-based method with prediction capability in case of insufficient features, relying on a point-wise motion model. Secondly, we employ a robust model-based hierarchical position localization method to handle change of image features along a range of distances, and localize an attitude-controlled (partially cooperative) satellite. Moreover, the thesis presents a pose tracking method addressing ambiguities in edge-matching, and a pose detection algorithm based on appearance model learning. For the validation of the methods, real camera images and ground truth data, generated with a laboratory tet bed similar to space conditions are used. The experimental results indicate that camera based methods provide robust and accurate tracking for the approach of malfunctioning satellites in spite of the difficulties associated with specularities and direct sunlight. Also exceptional lighting conditions associated to the sun angle are discussed, aimed at achieving fully reliable localization system in a certain mission.

<div>The orbits of interest for potential missions are stable or nearly stable to maintain long term presence for conducting scientific studies and to reduce the possibility of rapid departure. Near Rectilinear Halo Orbits (NRHOs) offer such stable or nearly stable orbits that are defined as part of the L1 and L2 halo orbit families in the circular restricted three-body problem. Within the Earth-Moon regime, the L1 and L2 NRHOs are proposed as long horizon trajectories for cislunar exploration missions, including NASA's upcoming Gateway mission. These stable or nearly stable orbits do not possess well-distinguished unstable and stable manifold structures. As a consequence, existing tools for stationkeeping and transfer trajectory design that exploit such underlying manifold structures are not reliable for orbits that are linearly stable. The current investigation focuses on leveraging stretching direction as an alternative for visualizing the flow of perturbations in the neighborhood of a reference trajectory. The information supplemented by the stretching directions are utilized to investigate the impact of maneuvers for two contrasting applications; the stationkeeping problem, where the goal is to maintain a spacecraft near a reference trajectory for a long period of time, and the transfer trajectory design application, where rapid departure and/or insertion is of concern.</div><div><br></div><div>Particularly, for the stationkeeping problem, a spacecraft incurs continuous deviations due to unmodeled forces and orbit determination errors in the complex multi-body dynamical regime. The flow dynamics in the region, using stretching directions, are utilized to identify appropriate maneuver and target locations to support a long lasting presence for the spacecraft near the desired path. The investigation reflects the impact of various factors on maneuver cost and boundedness. For orbits that are particularly sensitive to epoch time and possess distinct characteristics in the higher-fidelity ephemeris model compared to their CR3BP counterpart, an additional feedback control is applied for appropriate phasing. The effect of constraining maneuvers in a particular direction is also investigated for the 9:2 synodic resonant southern L2 NRHO, the current baseline for the Gateway mission. The stationkeeping strategy is applied to a range of L1 and L2 NRHOs, and validated in the higher-fidelity ephemeris model.</div><div><br></div><div>For missions with potential human presence, a rapid transfer between orbits of interest is a priority. The magnitude of the state variations along the maximum stretching direction is expected to grow rapidly and, therefore, offers information to depart from the orbit. Similarly, the maximum stretching in reverse time, enables arrival with a minimal maneuver magnitude. The impact of maneuvers in such sensitive directions is investigated. Further, enabling transfer design options to connect between two stable orbits. The transfer design strategy developed in this investigation is not restricted to a particular orbit but applicable to a broad range of stable and nearly stable orbits in the cislunar space, including the Distant Retrograde Orbit (DROs) and the Low Lunar Orbits (LLO) that are considered for potential missions. Examples for transfers linking a southern and a northern NRHO, a southern NRHO to a planar DRO, and a southern NRHO to a planar LLO are demonstrated.</div>